As we know, Krypto only joined Superboy in his teens, but that doesn't mean Superbaby didn't dream of his lost pup.
In this story, Baby Kal is reminded of having had a dog after being forbidden to play with other children lest he expose his super-powers. He also recalls the outlandish pets available on Krypton, even if he stuck to a simple dog.
After a restless night, Baby Kal asks his adoptive parents to get him a pet. They point to the farm animals they tend and tell little Clark he can pick one from the barn yard. Superbaby chooses the billy goat, ostensibly because a kid in his dream had a goat-like pet, though perhaps its white coat had something to do with it. But all Clark gets for his trouble is a belly full of leather and not much friendship at all.
Coincidentally, the Kents take Clark to the zoo, where he gets into trouble trying to wash the local gorilla, and breaking the hose nozzle in the process. That broken tube comes in handy later when Pa Kent has to explain how Baby Kal survived his encounter with the tiger he tried to domesticate.
The next day, Pa brings Superbaby to the ocean for a fishing trip, where the idea of a pet whale into his head.
After escaping the leviathan's maw, he accidentally frees a semi-intelligent "sea dragon" from a crevasse in the ocean floor and after following it back down into its lair, discovers a whole brood of them. But do they want to be his pets? Heck no. They want quite the opposite in fact.
That night, Superbaby follows a meteor and finds it to be a shapeshifting blob of protoplasm that reacts to his thoughts. The creature turns into various forms from his imagination, each one attacking viciously, except he can't tell because he's invulnerable. When he finally turns into a Kryptonian Battery Beast, Baby Kal finds he's a little too much to handle.
Water takes care of THAT threat, and apparently Baby Kal takes the creature off Earth, but that still leaves him petless. Until the next morning when Martha's rancher cousin Tom shows up with a dogie!
But not a doggie. Ha ha. You wanna bet the whole story started with that joke?
Meanwhile, in Adventure Comics #343, Krypto plays a small role defeating the Luck Lords with the Legion of Super-Pets.